The problem statement
Let $u\left( {x,y} \right),v\left( {x,y} \right)$ the solutions of the following problems:
$$\left\{ \begin{array}{l} \Delta u = {e^{{x^2} + {y^2}}}{\rm{ \quad\quad }}{x^2} + {y^2} < 1 \quad\quad\quad{\rm{ }}\\ {\left. u \right|_{{x^2} + {y^2} = 1}} = 1{\rm{ }} \end{array} \right.\left\{ \begin{array}{l} \Delta v = {e^{{x^2} + {y^2}}}{\rm{ \quad\quad }}\left| x \right| + \left| y \right| < 1{\rm{ }}\\ {\left. v \right|_{\left| x \right| + \left| y \right| = 1}} = 1{\rm{ }} \end{array} \right.$$ Prove that $u\left( {0,0} \right) < v\left( {0,0} \right)$
My attempt:
First notice that $u\left( {x,y} \right)$ is sub harmonic in the unit disk and thus satisfies the maximum principle. As $u\left( {x,y} \right)$ is obviously not constant we can conclude that $u\left( {x,y} \right) < 1$ in the unit disk.
Next we define $w\left( {x,y} \right) = u\left( {x,y} \right) - v\left( {x,y} \right)$ and notice that: $$\left\{ \matrix{ \Delta w = 0{\rm{ \quad \quad }}\left| x \right| + \left| y \right| < 1{\rm{ }} \hfill \cr {\left. w \right|_{\left| x \right| + \left| y \right| = 1}} = {\left. u \right|_{\left| x \right| + \left| y \right| = 1}} - 1{\rm{ }} \hfill \cr} \right.$$ From the previous paragraph we have that $${\left. w \right|_{\left| x \right| + \left| y \right| = 1}} = {\left. u \right|_{\left| x \right| + \left| y \right| = 1}} - 1 < 0{\rm{ }}$$ Since $w\left( {x,y} \right)$ is harmonic in $\left| x \right| + \left| y \right| < 1$ we get from the maximum principle: $$w\left( {0,0} \right) < {\max _{\left| x \right| + \left| y \right| = 1}}w\left( {x,y} \right) < 0$$ and in conclusion: $$u\left( {0,0} \right) < v\left( {0,0} \right)$$ $\blacksquare$
Is my solution correct? I'd appreciate any kind of feedback.